Non-conformal entanglement entropy

Journal of High Energy Physics, Jan 2018

We explore the behaviour of renormalized entanglement entropy in a variety of holographic models: non-conformal branes; the Witten model for QCD; UV conformal RG flows driven by explicit and spontaneous symmetry breaking and Schrödinger geometries. Focussing on slab entangling regions, we find that the renormalized entanglement entropy captures features of the previously defined entropic c-function but also captures deep IR behaviour that is not seen by the c-function. In particular, in theories with symmetry breaking, the renormalized entanglement entropy saturates for large entangling regions to values that are controlled by the symmetry breaking parameters.

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Non-conformal entanglement entropy

Implicitly this expression assumes that Non-conformal entanglement entropy Marika Taylor 0 1 William Woodhead 0 1 0 High eld , Southampton, SO17 1BJ , U.K 1 Mathematical Sciences and STAG Research Centre, University of Southampton We explore the behaviour of renormalized entanglement entropy in a variety of holographic models: non-conformal branes; the Witten model for QCD; UV conformal RG ows driven by explicit and spontaneous symmetry breaking and Schrodinger geometries. Focussing on slab entangling regions, we nd that the renormalized entanglement entropy captures features of the previously de ned entropic c-function but also captures deep IR behaviour that is not seen by the c-function. In particular, in theories with symmetry breaking, the renormalized entanglement entropy saturates for large entangling regions to values that are controlled by the symmetry breaking parameters. AdS-CFT Correspondence; Gauge-gravity correspondence - 4.1 4.2 2 Renormalized entanglement entropy 3 AdS entanglement entropy in general dimensions 4 Non-conformal branes Entanglement functional and surfaces Witten model 5 Renormalized entanglement entropy for RG ows 1 Introduction S = c1 D + c0 ln(R= ) + c~0; { 1 { Spontaneous symmetry breaking: Coulomb branch of N = 4 SYM 5.1.1 5.1.2 Coulomb branch disk distribution Coulomb branch spherical distribution 5.2 Operator driven RG ow 6 Non-relativistic deformations 7 Interpretation and comparison to QFT results 8 Conclusions and outlook 1 Introduction Entanglement entropy is widely used in condensed matter physics, quantum information theory and, more recently, in high energy physics and black holes. Consider a reduced density matrix A, obtained from tracing out certain degrees of freedom from a quantum system. The associated entanglement entropy is then the von Neumann entropy: S = Tr ( A ln A) : Throughout this paper we will be interested in the case for which a quantum system is subdivided into two, via partitioning space. In such a case A is a spatial region, with The entanglement entropy characterizes the nature of the quantum state of a system. For example, in the ground state of a quantum critical system in D spatial dimensions: where c1 D, c0 and c~0 are dimensionless; R is a characteristic scale of the region A and is an UV cuto . Logarithmic terms arise when D is odd, and their coe cients are related to the a anomalies of the stress energy tensor. More generally, the famous area law leading term characterizes the ground state of a system and can be used to test trial ground state wavefunctions. Entanglement entropy can also be used to distinguish between di erent phases of a system, such as the con ning/decon ning phase transition [1]. Continuum quantum eld theory (with a cuto ) is often used as a tool to describe discrete condensed matter systems. In this context, the cuto appearing in (1.2) is related to the underlying physical lattice scale in the discrete system and the coe cients of power law terms such as c1 D capture the leading physical contributions to the entanglement eld theory perspective, the expansion in (1.2) implicitly assumes the use of a direct energy cuto as a regulator. Di erent methods of regularisation result in di erent regulated divergences and thus the power law divergences are often called nonuniversal. By contrast logarithmic divergences are often denoted as universal as their coe cients are related to the anomalies of the theory. In even spatial dimensions, the logarithmic term in (1.2) is absent but the constant term c~0 is believed to be related to the number of degrees of freedom of the system. However, c~0 is manifestly dependent on the choice of the cuto . In two spatial dimensions, if (1.3) (1.4) (1.5) (1.6) for a spatial region with boundary of length R, then changing the cuto as R S = c 1 + c~0 ! 0 [2{4]. However, such an approach has several drawbacks. The regularisation is speci c to the shape of the geometry (a slab) and a modi ed prescription is needed for curved entangling region boundaries such as spheres, for which the scale of the entangling region is related to the local curvature of the entangling region boundary (see proposals in [5]). Any such prescription depends explicitly on the UV behaviour of the theory. More generally, extraction of nite terms by di erentiation obscures scheme dependence: there is no connection with the renormalization scheme used for other QFT quantities such as the partition function and correlation functions. From a quantum eld theory perspective, as opposed to a condensed matter perspective, it is very unnatural to work with a regulated rather than a renormalized quantity. In previous papers [6, 7], we introduced a systematic renormalization procedure for entanglement entropy, in which the counterterms are inherited directly from the partition function counterterms. As we review in section 2, such renormalization guarantees that the counterterms depend only on the quantum eld theory sources (non-normalizable modes in holographic gravity realisations) and not on the state of the quantum eld theory (normalizable modes in holographic gravity realisations). The renormalized entanglement entropy Sren expressed as a function of a characteristic scale of the entangling region implicitly captures the behaviour of the theory under an RG ow: small entangling regions probe the UV of the theory, while larger regions probe the IR. In this paper we will establish how these nite contributions to entanglement entropy behave in a variety of theories, using holographic models. The plan of this paper is as follows. In section 2 we review the de nition of renormalized entanglement entropy introduced in [6]. In section 3 we calculate the renormalized entanglement entropy for a slab region in anti-de Sitter (in general dimensions). The latter is relevant for the non-conformal branes discussed in section 4, as the latter can be viewed as dimensional reductions of anti-de Sitter theories in general dimensions. In section 4 we also compute the renormalized entanglement entropy for a slab region in the Witten holographic model for QCD. Section 5 explores renormalized entanglement entropy for operator and driven holographic RG ows, which are UV conformal. In section 6 we consider renormalized entanglement entropy in holographic Schrodinger geometries. In section 7 we summarise the main features of the renormalized entanglement entropy, using both our holographic results and earlier perturbative/lattice calculations. We conclude in section 8. 2 Renormalized entanglement entropy Entanglement entropy is usually calculated using the replica trick. The Renyi entropies where Z(1) is the partition function and Z(n) is the partition function on the replica space obtained by gluing n copies of the geometry together along the boundary of the entangling region. The entanglement entropy is obtained as the limit Note that this limit implicitly assumes that the Renyi entropies are analytic in n. Both sides of (2.1) are UV divergent. In a local quantum eld theory, the UV divergences of log Z(n) cancel with those of n log Z(1) except at the boundary of the entangling region; therefore the U V divergences of S(n) scale with the area of this boundary. S = nli!m1 Sn: { 3 { (2.1) (2.2) We can formally de ne the renormalized entanglement entropy as [6] Snren = 1 (1 n) (log Zren(n) n log Zren(1)) (2.3) with Sren = S1ren. Here the renormalized partition functions are de ned with any suitable choice of renormalization scheme. The replica space matches the original space, except at the boundary of the entangling region where there is a conical singularity. To de ne the renormalization on the replica space it is therefore natural to work within a renormalization method that works for generic curvature backgrounds for the quantum eld theory. is the UV cuto , Vd is the volume of the background (Euclidean) geometry, m2 is the mass and R is the Ricci scalar. The coe cients (ad; ad 2; bd 2; and in the above expressions we ignore boundaries of M. ) are dimensionless The divergences of the partition function on the replica space Z(n) have exactly the same structure and coe cients. However, the curvature of the replica space has an additional term from the conical singularity [8, 9] 1)2; where (@ ) is localised on a constant time hypersurface, on the boundary of the entangling region. (Here and in what follows we consider only static situations.) Therefore, when we use the replica formula (2.2) the leading divergences of the partition functions (scaling with the volume) cancel so that the leading divergent term in the entanglement entropy behaves as Sreg = 4 bd 2 d 2 d d 2xp + Such a divergence can clearly be cancelled by the counterterm Sct = 4 bd 2 d 2 d d 2xp ; which is covariantly expressed in terms of the geometry of the entangling region. Z 2.2 In gauge-gravity duality, the de ning relation is [ 10, 11 ] IE = log Z; (2.8) where IE is the onshell action for the bulk theory dual to the eld theory. In the supergravity limit this is given by the onshell Euclidean Einstein-Hilbert action together with appropriate matter terms i.e. IE = 1 Z d p d x h (K + ) ; (2.9) where the latter is the usual Gibbons-Hawking-York boundary term. The volume divergences of the bulk gravity action correspond to UV divergences of the dual quantum eld theory; these divergences can be removed by appropriately covariant counterterms at the conformal boundary. gravity the action counterterms are For example, in the case of asymptotically locally anti-de Sitter solutions of Einstein Ict = 1 Z d p d x h (d 1) + R 2(d 2) + where the ellipses denote terms of higher order in the curvature and logarithmic counterterms arise for d even. Applying the replica formula to the bulk terms in the action, as discussed in [12], and using the analogue of (2.5) for the bulk curvature, namely, (2.10) (2.11) (2.12) (2.13) gives the Ryu-Takayanagi functional [13] for the entanglement functional: Applying the replica formula to the counterterms gives p (1 + ) ; with the leading counterterm being proportional to the regulated area of the entangling surface boundary. Analogous expressions for higher derivative gravity and gravity coupled to scalars can be found in [6]. Using a radial cuto to regulate is perhaps the most geometrically natural way to renormalize the area of the minimal surface but it is not the only holographic renormalization scheme. Dimensional renormalization for holography was developed in [14] and this method could also be used to renormalize the holographic entanglement entropy. { 5 { AdS entanglement entropy in general dimensions In this section we review the renormalized entanglement entropy for a slab domain in Antide Sitter in general dimensions. The regulated entanglement entropy for such slab domains was analysed in [13]; here we extract from their analysis the renormalized entanglement entropy, in general dimensions. This quantity is relevant to the non-conformal brane backgrounds discussed in the next section, as the latter can be understood in terms of parent Anti-de Sitter theories, and also relevant for the Schrodinger backgrounds discussed in section 6. Let us parameterise AdSD+2 as counterterm is the regulated area of the boundary i.e. Z d D 1p~ h The entangling functional is We now consider an entangling region in the boundary of width L in the x direction, on a constant time hypersurface, longitudinal to the other (D 1) coordinates y . The bulk entangling surface is then speci ed by the hypersurface x( ) minimising where x0 = @ x. The equation of motion admits the rst integral where 0 is the turning point of the surface, related to L via or equivalently L = 2 0 xDdx x2D The regulated onshell value of the entangling functional is then Sreg = Vy 2GD+2 Dq d 1 2D Dd q 2D 0 = 0 (where we assume that D 6= 1) and therefore which can be rewritten in terms of dimensionless quantities as Dq d 1 2D 2D 0 (D (D 1 2 3 1 1) D 1 5 ; 1 1)~D 1 : This can be evaluated to give and hence As we discuss later, this quantity is closely related to the entropic c function for slabs in anti-de Sitter computed in [3]. In the case of D = 1 (AdS3) the entangling functional is logarithmically divergent, and the renormalized entanglement entropy depends explicitly on the renormalization scale: for a single interval Z d10xpGN e 2 2(8 1 p)!N 2 jF8 pj 2 (3.9) (3.10) (3.11) (3.12) (3.13) (4.1) (4.2) HJEP01(28)4 where the constants ( ; ) are given below for Dp-branes and fundamental strings respectively. (Note that it is convenient to express the eld strength magnetically, so for p < 3 we use Fp+2 = equations following from the action above can be reduced over a sphere, truncating to a (p + 2)-dimensional metric and scalar. The resulting action is then Id+1 = N Z dd+1xpge { 7 { where coupling as p = 2(p 4) 2 p 5 (5 9 p p+1 p) p 5 p 5 and gd2 is the dimensionful coupling of the dual eld theory, which is related to the string At any length scale l there is an e ective dimensionless coupling constant where d = p + 1 and the constants (N ; ; ; C) depend on the type of brane under consideration. For Dp-branes = 2 d 2 + dx dxd = (p 7)(p 3) 4(p 5) { 8 { For the fundamental string = 2 3 N = 3 gsN 2 ( 0)1=2 p and the dimensionful coupling is so AdSd+1 solution: In all cases, the dual frame is chosen such that the equations of motion admit an where the constant again depends on the case of interest: for Dp-branes (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) (4.11) HJEP01(28)4 = For further discussion of this point, see [16]. Let us de ne a parameter as The non-conformal branes are formally related to AdS gravity in the following way [17]. HJEP01(28)4 Now we consider (2 + 1)-dimensional gravity with cosmological constant = (2 1), I(2 +1) = NAdS Z d 2 +1xpg2 +1 (R2 +1 + 2 (2 1)) : so that the action is Reducing on a (2 ansatz results in the action (4.2) where d)-dimensional torus with coordinates za via a diagonal reduction ds2 = ds2d+1(x) + exp (2 2 d) dzadza while = 3=4 for fundamental strings. In general the equations admit an AdS solution with linear dilaton provided that the parameters are related as (4.12) (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) (4.19) N = NAdSV(2 d); with V(2 d) the volume of the compacti cation torus. 4.1 Entanglement functional and surfaces The entanglement functional follows from the replica trick: in the dual frame The equations for the entangling surface can be expressed geometrically as S = 4 N Z d d 1 p x he Km = where gmn is the background metric, hij is the induced metric on the entangling surface, Xm(xi) speci es the embedding of the entangling surface into the background and Km are the associated traces of the extrinsic curvatures. The dual frame entanglement functional follows directly from the reduction of the pure gravity entanglement functional S = 4 NAdS d The renormalized entanglement entropy for a strip in the F1 background can be exwhen one again uses the diagonal reduction ansatz (4.15), and assumes that the entangling surface wraps the torus and that the shape of the surface does not vary along the torus directions. In the upstairs picture the entangling surface satis es KM = 0; where the background metric is now denoted g(2 +1)MN and KM denotes the traces of the extrinsic curvatures. Thus, any AdS entangling surface which factorises as 2 1 = T (2 d) will give an entangling surface for non-conformal branes; moreover, the nonconformal brane surface will inherit its renormalized entanglement entropy from the upstairs entangling surface. entangling functional is As an example, let us consider slab entangling regions, characterised by a width x = L. The bulk entangling surface is speci ed as x( ) and in the background (4.10) the S = 4 N Vy Z d which is indeed precisely the functional obtained in (3.3), identifying D = (2 renormalized entanglement entropy can then be expressed as pressed as where (4.20) (4.21) (4.22) (4.23) (4.24) (4.25) (4.26) where the e ective coupling is expressed as ge2 (L) = gf2N L2. The expression for the renormalized entanglement entropy of a strip in the D1 background is analogous: 4.2 Witten model six-dimensional background: The Witten [18] holographic model for YM4 can be expressed in terms of the following Sren = Sren = Regularity of the geometry requires that the circle direction must have periodicity 2 3 L = KK : This model originates from D4-branes wrapping the circle with anti-periodic boundary conditions for the fermions. which breaks the supersymmetry. At low energies the model resembles a four-dimensional gauge theory, with the gauge coupling being g2 = g52=L . The gravity solution captures the behaviour of this theory in the limit of large 't Hooft coupling 2 = g2N and Sugimoto [19, 20] introduced D8-branes wrapped around the S4 on which the theory is reduced from ten to six dimensions. These D8-branes model chiral avours in the dual gauge theory and the resulting Witten-Sakai-Sugimoto model has been used extensively as a simple holographic model of a non-supersymmetric gauge theory with avours. The operator content of the dual theory captured by the metric and scalar eld is the four-dimensional stress energy tensor Tab, a scalar operator O component of the ve-dimensional stress energy tensor T corresponding to the and the gluon operator O corresponding to the bulk scalar eld. These operators satisfy a Ward identity [16] One of the main applications of this model is in the context of avour physics: Sakai and their expectation values can be extracted from the above geometry. For example, the condensate of the gluon operator hTaai + hO i + 1 g2 hOi = 0 hOi = 2 5 2 2N 37 L4 and therefore L controls the QCD scale of the theory. Next we can consider a slab entangling region, wrapping the circle direction , characterised by a width x = L. Entanglement entropy in this theory was previously discussed in [1], with the con nement transition being associated with a discontinuity in the derivative of the entanglement entropy with respect to L. The bare entanglement functional is S = 4 N V2L 5 Z d p1 + f ( )(x0)2 where V2 is the volume of the two-dimensional cross-section of the slab. The entanglement can then be written as Sreg = 8 N V2L Z d pf ( ) 10 5 pf ( ) 10 f ( ) 10 where is the turning point of the surface, related to the width of the entangling region as L = 2 Z r f ( ) d 10f( ) 10f( ) 1 : 0.0 0.1 solid orange lines indicate the renormalized entropy for the two possible connected minimal surfaces. The entanglement entropy can be renormalized as before, with the counterterm contributions being Sct = 2 N V2 4 : L For large entangling regions, the only possible entangling surface is the disconnected conguration, for which the renormalized entanglement entropy is Sren = 8 N V2L = For small entangling regions the condensate is negligible and the renormalized entanglement entropy is controlled by the conformal structure The renormalized entanglement entropy is plotted in gure 1. As discussed in [1] there is a discontinuity in the derivative of the entanglement entropy for slab widths around L 0:4 KK . For larger values of L the entanglement entropy saturates at a constant value. Renormalized entanglement entropy for RG ows In this section we will consider holographic entanglement entropy in geometries dual to RG ows. We work in Euclidean signature with a bulk action I = 1 1 2 Holographic RG ows with at radial slices can be expressed as ds2 = dr2 + exp(2A(r))dxidxi; where the warp factor A(r) is related to a radial scalar eld pro le (r) via the equations of motion d 2 dr2 + d dA dr = dV d d2A dr2 = 1 d 2(d 1) dr 2 : These equations can always be expressed as rst order equations [21] where the (fake) superpotential W ( ) is related to the potential as d dr dA dr d dr = W = 2(d 1) dW d V = (d 1)2 2 dW d 2 d d 1 ! W 2 : S = Vy Z 4G dre(D 1)A(r)q 1 + e2A(r)(x0)2; (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) (5.9) Near the conformal boundary the potential can be expanded in powers of the scalar eld as and hence the superpotential can be written as V = d(d 1) = d=2 + pd2 + 4m2=2. The higher order terms in the superpotential are not unique, as di erent choices are associated with di erent RG ows. Note that for at domain walls, a single counterterm (in addition to the usual Gibbons Hawking term) is su cient Ict = (d 1) Z although the derivation of the entanglement entropy counterterms requires knowledge of the counterterms for a curved background (since the replica space is curved). The entanglement entropy for a slab region x = L in the RG ow geometry is where D is the number of spatial directions in the dual theory, Vy is the regulated volume of the longitudinal directions and x(r) de nes the entangling surface. Then where at the turning point r0 of the surface A(r) = A0. The regulated onshell action is with the cuto being r = . The entanglement entropy counterterms for RG ows driven by relevant deformations were discussed in [6], working perturbatively in the deformation. Here we will analyse both spontaneous and explicit symmetry breaking, using exact supergravity solutions. Spontaneous symmetry breaking: Coulomb branch of N = 4 SYM In this section we consider the case of VEV driven ow, i.e. spontaneous symmetry breaking. In such a situation, the scalar eld has only normalizable modes and thus asymptotically the scalar eld behaves as where (0) is related to the operator expectation value as L = 2 Z 1 eDA0 dr r0 eA(r)pe2DA(r) e2DA0 dr e(2D 1)A(r) pe2DA(r) e2DA0 e(d 2) ! (d 2) : (5.10) (5.11) (5.12) (5.13) (5.14) (5.15) (5.16) (5.17) From (5.4) and (5.7), one can immediately read o the asymptotic form of the warp factor: Substituting into the regulated action, we then obtain 2)=2, and is logarithmically divergent for = (d 2)=2. (The latter case does not however arise holographically, as when the lower bound on the conformal dimension is saturated the operator automatically obeys free eld equations.) Therefore, for VEV driven ows the only counterterm required is the regulated area of the boundary of the entangling surface: Sct = 1 Z 4(d d 2xph: Note that one can derive the same result from the bulk action counterterms, using the replica trick; see below for the case of the Coulomb branch of N = 4 SYM. Thus the renormalized entanglement entropy for slabs in VEV driven ows is Now let us consider the general structure of the renormalized entropy. In the vacuum of the conformal eld theory, the renormalized entropy must behave as with c0 a dimensionless constant on dimensional grounds: the entropy scales with the longitudinal volume Vy and the width of the entangling region L is the only other dimensionful scale in the problem. The value of c0 in holographic theories is given in (3.12). Now working perturbatively in the operator expectation value hOi the renormalized entropy must behave as where c1 is dimensionless and we work in a limit in which (5.18) (5.19) (5.20) (5.21) (5.22) (5.23) (5.24) (5.25) i.e. the width of the entangling region is much smaller than the length scale set by the condensate. 5.1.1 Coulomb branch disk distribution We now analyse a speci c example: the renormalized entanglement entropy of slab domains on the Coulomb branch of N = 4 SYM. We consider the case of a disk distribution of branes preserving SO(4) SO(2) symmetry, for which the equations of motion follow from (5.1), with the superpotential being [22] Using the standard Fe erman-Graham coordinates near the conformal boundary: The metric in ve-dimensional gauged supergravity is then p 6 1 3 4 p 6 ds2 = 2w2 dw2 Here the coordinate w ! 1 at the conformal boundary and characterises the expectation value of the dual scalar operator. The scalar eld can be expressed by the relation = p 2 2 + 1 2 1 6 dx dx operator, following [23, 24]: where we use the standard relation between the Newton constant and the rank of the dual The vanishing of the dual stress energy tensor is required given the supersymmetry but careful holographic renormalization is required to derive this answer. The regulated entanglement entropy of a slab domain in this geometry can be written as HJEP01(28)4 Using the rst integral of the equations of motion the width of the entangling region can be expressed in terms of the turning point of the surface w0 as Sct = V2 8G5 (5.26) (5.27) (5.28) (5.29) (5.30) (5.31) (5.32) (5.33) (5.34) where c is an integration constant and w0 satis es The regulated entanglement entropy is then L = 2 cdw w0 w2 3p 6w6 c2 and the required counterterm is expressed in terms of the regulated area of the boundary of the entangling surface i.e. there are counterterm contributions at each side of the slab. (The total contribution is therefore twice this value.) Note that the counterterms in this case clearly contribute both divergent and nite parts: expanding in powers of the cuto It is then convenient to write the entanglement entropy in terms of dimensionless quantities as lim !1 Z ~ y0 p y y + 1 dy py2(y + 1) and y0 is the turning point of the surface. Then L = y0py0 + 1 y0 yp(y + 1)py2(y + 1) These integrals can be computed numerically. There is a maximal value of L (for xed ) for which a connected entangling surface exists: the critical value of L is such that 1 3 : 2 e p 2 6 1 e p 6 = 2w2: The metric in ve-dimensional gauged supergravity is then with 2 3 2 p 6 p 6 ds2 = 2w2 dw2 Coulomb branch spherical distribution We now consider the renormalized entanglement entropy of slab domains on the Coulomb branch of N = 4 SYM for the case of a spherical distribution of branes, preserving SO(4) SO(2) symmetry. The equations of motion follow from (5.1), with the superpotential being [22] Here the coordinate w ! 1 at the conformal boundary and characterises the expectation value of the dual scalar operator. The scalar eld can be expressed by the relation Lcrit 1:5708: Sren = 12G5 : For L > Lcrit there is no connected entangling surface but the disconnected entangling surface consisting of two components x = L=2 and x = L=2 still exists. For the latter one can straightforwardly calculate the renormalized entanglement entropy as HJEP01(28)4 The renormalized entanglement entropy is plotted in gure 2: its rst derivative is discontinuous at L = Lcrit. For small values of L, the analytic expressions (5.19) is valid: Sren = V2 G5 0 3 2 !2 1 ( 1 ) 6 and the constant C1 can be determined as: C1 0:03137: (5.35) (5.36) (5.37) (5.38) (5.39) (5.40) (5.41) (5.42) (5.43) -2 -4 -6 -8 -10 the numerical results for a cut-o of = 1010, the dotted red line shows the small L t of equation (5.38), the dashed yellow line shows the value of the renormalized entropy for disconnected surfaces. Using the standard Fe erman-Graham coordinates near the conformal boundary: ds2 = d 2 + = p 2 2 + 1 2 1 6 1 2 1 18 We can then read o the expectation values of the dual stress energy tensor and scalar operator, following [23, 24]: = where we use the standard relation between the Newton constant and the rank of the dual The vanishing of the dual stress energy tensor is required given the supersymmetry but again careful holographic renormalization is required to derive this answer. The regulated entanglement entropy is then Sreg = V2 Z and the required counterterm is expressed in terms of the regulated area of the boundary of the entangling surface i.e. there are counterterm contributions Sct = V2 8G5 (5.44) (5.45) (5.46) (5.47) (5.48) the counterterms in this case clearly contribute both divergent and nite parts: expanding Sct = It is then convenient to write the entanglement entropy in terms of dimensionless quantities as where ~ is a rescaled dimensionless cuto . Implicitly this expression assumes that 2 6= 0 and y0 is the turning point of the surface. Then L = y0py0 1 Z 1 dy y0 yp(y 1)py2(y 1) y02(y0 1) These integrals can again be computed numerically. As in the previous case, for xed there is a maximal value of L for which a connected entangling surface exists. The critical value is For lengths grater than the critical length, the minimal surface is disconnected and the renormalized entanglement entropy can be calculated analytically to give Lcrit 0:8317 Sren = V2 2 6G5 : Lc 0:75 (5.49) (5.50) (5.51) (5.52) (5.53) (5.54) (5.55) (5.56) For subcritical values, there are two possible surfaces with turning points y0 for each width L and one must choose the surface for which the renormalized area is minimised. The renormalized entanglement entropy is plotted in gure 3. There is a phase transition between the connected and disconnected entangling surfaces at Lc such that i.e. Lc < Lcrit, and the entanglement entropy is saturated for L Lc. In the regime of small L the analytic expressions (5.19) are valid: where the constant C1 can be determined as: C1 0:03167 -0.5 -1.0 -1.5 -2.0 -2.5 solid orange lines indicate the renormalized entanglement entropy for the two possible connected minimal surfaces. The dashed orange line indicates the entanglement entropy for the disconnected surface. The dotted red line shows the small L t of equation (5.55). In this section we consider the case of an operator driven RG ow, the GPPZ ow [25]. The equations of motion again follow from (5.1), with the superpotential being The metric can be expressed as while the scalar eld is given by 1 2 p 3 ds2 = d 2 2 + 1 2 (1 2 2)dx dx The scalar is dual to a dimension three operator. By expanding near the conformal boundary and using the holographic renormalization dictionary, [23, 24] showed that the GPPZ solution is dual to a deformation (proportional to ) of N = 4 SYM by the dimension three scalar operator, with the expectation values of the operators being The vanishing stress energy tensor is again required by supersymmetry while the vanishing of the scalar VEV re ects the explicit (as opposed to spontaneous) symmetry breaking. = p 2 3 log 1 + 1 hTij i = hOi = 0: (5.57) (5.58) (5.59) (5.60) Now let us consider the renormalized entanglement entropy of a strip region in this geometry. The entanglement entropy can be expressed as The overall dependence on the deformation can be scaled out to give S = V2 Z 2G5 d (1 S = V2 2 Z 2G5 dv (1 3 v3 where v = and X = x. Then the entangling surface of width L satis es HJEP01(28)4 L = 2 Z v0 0 dv q v 3 (1 v2)4 v6 2(1 v2) where the integration constant is related to the turning point of the surface v0 by (5.61) (5.62) (5.63) (5.64) (5.65) (5.66) (5.67) (5.68) (5.69) = (1 v2)3=2 0 v 3 0 : Lcrit 0:3008: As in the previous cases there is a phase transition between a connected solution for L < Lcrit and a disconnected solution for L > Lcrit where The regulated entanglement entropy is then Sreg = V2 2 Z v0 2G5 dv (1 q v3 (1 v2)5=2 v2)3 v6 2 : Sct = d x 2 p~ h 1 + 3 2 2 log( ) The counterterms for the entanglement entropy can be derived from the bulk action counterterms using the replica trick: where the cuto in the coordinates is ~ = = . Evaluating this counterterm gives a contribution from each endpoint of the strip: Sct = V2 8G5 1 2 1 + 2 log ; which indeed matches the regulated divergences of (5.66). Thus the total renormalized entropy is Sren = V2 2 0 Z v0 dv (1 q v3 (1 v2)5=2 v2)3 v6 2 1 2 2 + 1 2 1 log A : 0.10 0.15 -0.5 -1.0 -1.5 -2.0 -2.5 ow. The solid blue line and solid orange lines indicate the renormalized entropy for the two possible connected minimal surfaces. Note that Sren > 0 near Lcrit for the connected solutions. These integrals can once again be evaluated numerically, the results of which are plotted in gure 4. As in the case of the spherical brane distribution, there are two possible turning points for a given length L < Lcrit, the branch with v0 < v0;crit is favoured for all such L. Both branches are positive near L = Lcrit, whereas it can be shown analytically that the renormalized entropy is zero for disconnected surfaces and so there is a transition from the connected to disconnected surface solutions at around Lc 0:27 (5.70) where the entanglement entropy has a discontinuous derivative. 6 Non-relativistic deformations Entanglement entropy is a natural computable in non-relativistic quantum eld theories that are used to describe condensed matter systems. As for relativistic systems, entanglement entropy can be used to classify di erent phases of the system, and the behaviour of the entanglement entropy as one scales up or down the entangling region size can elucidate the UV and IR behaviour of the theory. Holography provides a large class of duals to strongly interacting non-relativistic systems, and it is interesting to explore how entanglement entropy in these systems di ers from that in perturbative realisations of non-relativistic quantum eld theories. In this section we will explore Schrodinger geometries, which indeed exhibit an interesting transition from UV to IR behaviour. It would be interesting to explore renormalized entanglement entropy in Lifshitz geometries also, but non-trivial behaviour in this case requires time dependent setups | minimal surfaces at constant time in Lifshitz behave precisely as those in AdS. We therefore postpone exploration of (time dependent) renormalized entanglement entropy in Lifshitz for future work. Schrodinger metrics in (p + 3) dimensions can be written as [26] 2 1 ds2 = r2z (dx+)2 + r2 dr2 + dx+dx + dxidxi where the index i runs over p directions. The light cone coordinates can be rewritten as x = (y t) The metric can be supported by (real) massive gauge elds provided that b2 > 0 for z < 1 and b2 < 0 for z > 1. In both cases the dual eld theory can be understood as a deformation of the CFT by an operator that breaks the relativistic symmetry but respects non-relativistic scaling invariance i.e. the dual theory has the form [27] ICF T + Z dp+2x jbjO + where the operator O is a vector (or tensor) that picks out the x+ direction. The deformation is relevant (dimension less than (p + 2)) with respect to the original conformal symmetry for z < 1 and irrelevant for z > 1. Let us consider the case of z < 1, so that the theory remains UV conformal; this case was explored in detail in [28]. We can then specify a spacelike entangling region in the dual eld theory, at constant t, de ned by y(xi). (For z > 1, the situation is more complicated as surfaces of constant t are not spacelike at in nity and we will not discuss this case further here.) We can illustrate the behaviour of the entangling surfaces by two cases: a slab of width L in the y direction and a slab of width L along one of the xi directions. In the latter case the entangling surface in the bulk is described by w(r) (where w is the direction transverse to the entangling region) and the entangling functional is S1 = RyVp 1 Z 2G 1 r(p+1) p 1 + b2r2(1 z)p1 + w0(r)2dr where now Vp is the regulated volume of the xi directions. Note that both entangling functionals can be expressed in the form f (r)p1 + g(r)w0(r)2dr tional is S2 = Vp Z 2G 1 r(p+1) q 1 + (1 + b2r2(1 z))y0(r)2dr where Ry is the regulated length of the y direction and Vp 1 is the regulated volume of the In the other case the entangling surface is described by y(r) and the entangling func(6.1) (6.2) (6.3) (6.4) (6.5) (6.6) HJEP01(28)4 for suitable choices of the overall normalisation N and the functions (f (r); g(r)). Then the width of the entangling region is given by where ro is the turning point of the minimal surface and L = 2 Z 1 ro dr pg(r)(g(r)f (r)2 g(ro)f (ro)2) S = N Z 1 ro f (r)2pg(r) pg(r)f (r)2 g(ro)f (ro)2 dr: When f (r) and g(r) are monomials of r, the renormalized entanglement entropy can be calculated exactly using the AdSD result derived in section 3. For the slab along the xi directions, the renormalized entanglement entropy interpolates between the AdSp+3 result (for small slab widths) HJEP01(28)4 and the following result for large slab widths (S1)ren = RyVp 1 0 2p 0 2p 2(p+z) + 12 1p+z 1 1 2(p+z) A The latter expression applies for bL1 z 1, in which case the functional is approximated by S1 = RyVp 1 Z 2G b which is precisely the functional analysed in section 3 (taking D = p + z). In the other case, the renormalized entanglement entropy is also given by the AdSp+3 result for small slab widths while at large slab widths bL1 z 1 the relevant functional is S2 = Vp Z 2G 1 r(p+1) q 1 + b2r2(1 z)y0(r)2dr: Consider rst the special case of z = 0. By a change of variable we can express this functional as S2 = bVp Z 2G w b 1p p1 + y_(w)2dw; where y_ = dy=dw. From the general result (3.12), we can now read o the renormalized entanglement entropy as bVp 0 2p b 2p A (6.7) (6.8) (6.9) (6.10) (6.11) (6.12) (6.13) (6.14) which is manifestly consistent with (6.9) for b = z = 1. Thus the renormalized entanglement entropy scales di erently for large slab widths (such that bL1 z 1), depending on the orientation of the slab with respect to the y direction along which the theory is deformed away from conformality. In this case the explicit symmetry breaking is associated with a breaking of the relativistic symmetry, while preserving non-relativistic scale invariance, and the renormalized entanglement entropy does not have a discontinuity in its derivative and does not saturate in the deep IR. (Note however that the Schrodinger geometry has a null curvature singularity and thus quantum corrections to the geometry may change the deep IR behaviour.) 7 Interpretation and comparison to QFT results In the previous sections we have explored the renormalized entanglement entropy for slab domains in a variety of holographic models. While the general method of area renormalization is applicable to entangling domains of any shape, it is particularly convenient to use slab domains for several reasons. Firstly, the equations of motion admit rst integrals, thus simplifying the analysis. Secondly, slab entangling domains have been analysed for a variety of quantum eld theories in the literature. Note that the previous literature does not compute the renormalized entanglement entropy, but typically extracts instead As we discussed in the introduction, in any local quantum eld theory the divergences in the entanglement entropy are necessarily independent of the width of the slab, L, and thus c(L) is manifestly UV nite. Consider now the renormalized entanglement entropy. The counterterms include nite contributions, as illustrated in the previous section, but these nite contributions are independent of the width of the slab, as the counterterms are expressed in terms of local quantities at the boundaries of the entangling region. Therefore bVp For 0 < z < 1, the functional (6.12) can be expressed as and thus from (3.12) S2 = bVp 2G 1 zb p1 + y_(w)2dw pz +1 Z dw w pz +1 pz +1 0 2p and thus the slope of our renormalized quantity matches the c function de ned in earlier literature. This statement can be expressed as Z Sren = c(L)dL + s0 where s0 is independent of L, but dependent on parameters of the theory. Thus the renormalized quantity is an integrated version of the c function. (Note that the c(L) is de ned in various di erent ways in the literature. For example, [3] use a de nition of the entropic c function that incorporates factors of L.) Next let us consider the UV and IR behaviour of the renormalized entanglement entropy. The renormalized entanglement entropy measures the residual entanglement between the entangling region and its complement, after subtracting the divergent contributions arising from entanglement at the boundary. In the ground state of a conformal eld theory, correlation functions are characterised by power law behaviour and thus it would be reasonable to expect that the residual entanglement scales inversely with the width of the slab entangling region (and extensively with the length of the slab region). This heuristic argument is in agreement with the explicit holographic result (3.12). For a slab region in the ground state of a conformal eld theory, the L independent contribution s0 in (7.3) is necessarily zero, as there is no dimensionless ratio that is independent of L. The renormalized entanglement entropy is thus determined entirely by the c function, with the positivity of the latter implying the negativity of the former. For non-conformal branes, the entanglement entropy is controlled by the conformal structure in (d 2 ) dimensions, and therefore similar arguments apply. Suppose that in the IR of the theory correlation functions fall o exponentially with characteristic mass scale ; entanglement is thus signi cant only on length scales of order 1 from the entangling region boundaries. If the width of the entangling region L is much greater than this length scale, then we would expect the renormalized entanglement entropy to saturate to a value that is independent of L. On dimensional grounds this residual entanglement entropy must then scale for a d-dimensional theory as Vd 2 d 2 for a slab region of area Vd 2. Thus there is a non-vanishing constant term in (7.3) which would not be seen in the c function (which is in such cases zero for large L). This behaviour can be seen in a number of explicit QFT calculations. In [29] the entanglement entropy for massive scalar elds in various dimensions was computed, and expressed in term of the derivative of the entanglement entropy with respect to the mass . The latter is sensitive to the contributions to the renormalized entanglement entropy that are independent of L, and hence are lost from the c function. For example, for d = 3, it was shown that V1 24 with V1 1 the regulated length of the slab region. Integrating this expression results in a nite contribution to the renormalized entanglement entropy in agreement with the above arguments. In d = 4 the analogous expression is S S (7.4) (7.5) (7.6) Sren = V1 ; 12 V2 2 48 : These terms arise from logarithmic divergences in the regulated entanglement entropy V2 48 S = 2 : Whenever there is a logarithmic divergence, the renormalized entanglement entropy has scheme dependence, corresponding to the choice of nite counterterms [6]. Such logarithmic divergences occur in particular for CFTs in d dimensions deformed by operators of dimension = d=2 + 1 [30, 31]. The logarithmic divergences are removed by counterterms of the form in holographic realisations, where is the scalar eld dual to the deforming operator. By rescaling ! e this counterterm will change to (7.7) (7.9) (7.10) (7.8) d 2xp 2 log d d 2xp 2(log + ) with the latter term being nite (due to the operator dimension). In particular, using the leading asymptotic behaviour for the scalar eld, the latter term contributes a term Vd 2 s2 to the renormalized entanglement entropy, where s is the operator source. Therefore, the renormalized entanglement entropy depends explicitly on the choice of the coe cient of the nite term . (More generally, operators of dimension = d(1 1=2n)+1=n are associated with logarithmic divergences [6] and hence lead to nite terms in the entanglement entropy behaving as Vd 2 s2n.) In supersymmetric theories, the ambiguity can be xed by requiring that the renormalization scheme for the partition function respects supersymmetry and then using the replica trick to derive the counterterms for the entanglement entropy. The renormalization scheme for GPPZ, which indeed corresponds to a CFT deformed by a supersymmetric operator of dimension = d=2 + 1, was constructed to respect supersymmetry [23, 24]. It is thus perhaps unsurprising that the supersymmetric scheme implies that the renormalized entanglement entropy in this case vanishes in the deep IR. The discontinuity in the derivative of the entanglement entropy with respect to L in a holographic con ning theory was rst described in [1]. In the examples of explicit and spontaneous symmetry discussed here the renormalized entanglement entropy always saturates in the IR, and there is a discontinuity in the derivative of the entanglement entropy at the critical value of L, at which the dominant entangling surface becomes disconnected. Note however that the slope of the derivative can be small close to the transition point, as in one of our Coulomb branch examples, and one thus needs to ensure that the numerical resolution is su cient to capture the discontinuity in the derivative. In addition to calculations of the entanglement entropy in free eld theories, various calculations of the entanglement entropy for slab regions have been carried out in lattice gauge theories. In [ 32 ] the entanglement entropy for a slab of width L in a four-dimensional SU(2) gauge theory was studied numerically. The results of this study are in agreement with the behaviour found here. The derivative of the entanglement entropy with respect to L has a discontinuity at a critical value, as found in holographic con ning theories in [1] and discussed above, and it was also observed that there are nite contributions to the entanglement entropy which scale as 1=L2 for small width entangling regions. While [ 32 ] did not extract the renormalized entanglement entropy, their results imply that the renormalized entanglement entropy would scale as 1=L2 for small width entangling regions. The SU(2) gauge theory is asymptotically free and thus one would expect the renormalized entanglement entropy for small regions to be captured by free gluons, which indeed scales in accordance with the conformal result discussed earlier in the paper. Note that the residual nite contributions at large L were not computed in [ 32 ]. A more recent lattice simulation [ 33 ] studied entanglement entropy for slab regions in SU(3) gauge theory in four dimensions. The generic features are similar to those found in the SU(2) theory (free at small distances, c(L) goes to zero at nite L), although the detailed features near the critical length di er between SU(2) and SU(3). In particular, c(L) seems to go smoothly to zero at the critical length, and therefore there is no discontinuity in the derivative of the entanglement entropy with respect to L. As in [ 32 ], only the vanishing of the derivative of the renormalized entanglement entropy for large L was shown; the residual nite entanglement entropy was not computed. 8 Conclusions and outlook In this paper we have explored renormalized entanglement entropy for slab domains, for a variety of di erent holographic theories. We have shown that the renormalized entanglement entropy captures not just the features of the previously discussed entangling c function, but also the deep IR behaviour of symmetry breaking theories (where the c function vanishes). It would be interesting to analyse the properties of renormalized entanglement entropy for other common entangling regions, such as spheres and hypercubes. Note however that the latter are considerably more complicated to compute holographically: the equations of motion for the minimal surfaces do not admit rst integrals and the vertices of hypercubes are generally associated with additional logarithmic counterterms in the entanglement entropy. The examples discussed in this paper indicate the existence of general bounds on the renormalized entanglement entropy: Sren 0 with Sren ! 0 for supersymmetric RG ows triggered by operator deformations. It would be interesting to develop proofs of these bounds in future work. Related bounds were discussed in [34], although the functional analysed in [34] is not identical to the renormalized entanglement entropy considered here. Note that there are heuristic arguments why Sren 0. For CFTs in odd dimensions, following [35], the renormalized entanglement entropy for spherical regions is related to the partition function on a sphere, and the negativity of the renormalized entanglement entropy is thus related to the conjectured positivity of the F quantity [36]. (Away from the xed points, along the RG ow, the relationship between the F quantity (the free energy on the sphere) and the renormalized entanglement entropy is more complicated than the relation in [35] but nonetheless in all explicit examples positivity of F indeed maps to negativity of the renormalized entanglement entropy for a disk region.) More generally, the renormalized entanglement entropy coincides with minus the (renormalized) Euclidean action for a D(d 1)-brane with no worldvolume gauge elds and no Chern-Simons couplings to background uxes i.e. the latter is also a minimal surface. The (renormalized) Euclidean action is positive semi-de nite for stable D-brane embeddings, and vanishes for supersymmetric D-brane embeddings. This heuristic argument suggests that the renormalized entanglement entropy should be negative semi-de nite but does not however explain why the renormalized entanglement entropy is zero in the IR for supersymmetric operator driven ows but not for supersymmetric VEV driven ows. Holography allows us to explore entanglement entropy for a wide variety of strongly coupled quantum eld theories. In this work we have extracted from existing perturbative and lattice results the behaviour of the renormalized entanglement entropy for slabs but it would clearly be interesting to explore renormalized entanglement entropy directly within perturbative quantum eld theory, using varied renormalization methods. The replica trick allows us to derive the counterterms for the entanglement entropy but it would be useful to understand the role of these counterterms in computations of renormalized entanglement entropy via twist eld correlators. There has been considerable progress in understanding the computation of entanglement entropy in lattice gauge theories, see for example [32, 33, 37{39], and it would be interesting to explore how the continuum limit of such computations can be matched with our de nition of renormalized entanglement entropy. More generally, one would hope that it may become possible to calculate entanglement entropy for certain supersymmetric theories on the lattice in the near future | see for example [40] for recent progress on simulating N = 4 SYM. We can rewrite the holographic result (3.12) for the renormalized entanglement entropy for a slab in N = 4 SYM as Conformal invariance implies that the renormalized entanglement entropy has a leading behaviour at large N Sren 0:114 Sren = f (gY2MN ) N 2Vy L2 : N 2Vy L2 (8.1) (8.2) where f (gY2MN ) is a positive function of the 't Hooft coupling; it is this function that one would like to compute perturbatively using lattice simulations. One can estimate the free eld value of this function by summing contributions from the six real scalars, four Weyl fermions (equivalent to two Dirac fermions) and the gauge eld of N = 4 SYM. Estimating the gauge eld contributions by scaling the recent SU(3) result of [ 33 ] and taking the other contributions from [3, 41] we obtain f 0:05 at zero coupling. This suggests that the magnitude of f increases with the 't Hooft coupling, as one might expect. Finally, we would like to turn to issues of measurability. Throughout this paper we have focussed on the de nition and calculation of renormalized entanglement entropy in a UV complete quantum eld theory. We believe that this is an important computable: our systematic renormalization procedure makes it clear when the renormalized quantity is scheme independent and thus when nite residual terms are meaningful. The systematic renormalization also allows us to compare di erent phases. One may however be interested in a system for which only a low energy e ective theory description is known; this e ective eld theory may not be renormalizable. In such a context, one would rst calculate the regulated entanglement entropy in terms of the UV cuto for the system. If the e ective eld theory description is associated with a renormalizable eld theory, one could follow the procedures of this paper to de ne renormalized entanglement entropy. (This will indeed be the case if the IR theory is a CFT.) If however the e ective eld theory is not renormalizable, one will inevitably need to retain the cuto dependence in the entanglement entropy and work with the regulated quantity. In the latter context, one will not be able to extract in a meaningful way nite contributions to the entanglement entropy. Thus the renormalized entanglement entropy, as with other renormalized QFT quantities, is applicable to UV complete renormalizable quantum eld theories. Acknowledgments We would like to thank Antonio Rago for useful comments regarding lattice calculations of entanglement entropy. This work was supported by the Science and Technology Facilities Council (Consolidated Grant \Exploring the Limits of the Standard Model and Beyond"). We thank the Simons Center and the GGI for partial support during the completion of this work. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 690575. Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. (2007) 090 [hep-th/0611035] [INSPIRE]. (2009) 504005 [arXiv:0905.4013] [INSPIRE]. [1] I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of con nement, Nucl. Phys. B 796 (2008) 274 [arXiv:0709.2140] [INSPIRE]. Nucl. Phys. B 764 (2007) 183 [hep-th/0606256] [INSPIRE]. [2] H. Casini and M. Huerta, Universal terms for the entanglement entropy in 2 + 1 dimensions, [3] T. Nishioka and T. Takayanagi, AdS bubbles, entropy and closed string tachyons, JHEP 01 [4] P. Calabrese and J. Cardy, Entanglement entropy and conformal eld theory, J. Phys. A 42 [5] H. Liu and M. Mezei, A re nement of entanglement entropy and the number of degrees of freedom, JHEP 04 (2013) 162 [arXiv:1202.2070] [INSPIRE]. [6] M. Taylor and W. Woodhead, Renormalized entanglement entropy, JHEP 08 (2016) 165 [arXiv:1604.06808] [INSPIRE]. [7] M. Taylor and W. Woodhead, The holographic F theorem, arXiv:1604.06809 [INSPIRE]. presence of conical defects, Phys. Rev. D 52 (1995) 2133 [hep-th/9501127] [INSPIRE]. [9] S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [INSPIRE]. [10] S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE]. HJEP01(28)4 JHEP 01 (1999) 003 [hep-th/9807137] [INSPIRE]. JHEP 09 (2008) 094 [arXiv:0807.3324] [INSPIRE]. 04 (2009) 062 [arXiv:0901.1487] [INSPIRE]. [13] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE]. [14] A. Bzowski, Dimensional renormalization in AdS/CFT, arXiv:1612.03915 [INSPIRE]. [15] H.J. Boonstra, K. Skenderis and P.K. Townsend, The domain wall/QFT correspondence, [16] I. Kanitscheider, K. Skenderis and M. Taylor, Precision holography for non-conformal branes, [17] I. Kanitscheider and K. Skenderis, Universal hydrodynamics of non-conformal branes, JHEP [18] E. Witten, Anti-de Sitter space, thermal phase transition and con nement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE]. [19] T. Sakai and S. Sugimoto, Low energy hadron physics in holographic QCD, Prog. Theor. Phys. 113 (2005) 843 [hep-th/0412141] [INSPIRE]. (2005) 1083 [hep-th/0507073] [INSPIRE]. [20] T. Sakai and S. Sugimoto, More on a holographic dual of QCD, Prog. Theor. Phys. 114 041 [hep-th/0105276] [INSPIRE]. 631 (2002) 159 [hep-th/0112119] [INSPIRE]. [21] D.Z. Freedman, C. Nun~ez, M. Schnabl and K. Skenderis, Fake supergravity and domain wall stability, Phys. Rev. D 69 (2004) 104027 [hep-th/0312055] [INSPIRE]. [22] D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Continuous distributions of D3-branes and gauged supergravity, JHEP 07 (2000) 038 [hep-th/9906194] [INSPIRE]. [23] M. Bianchi, D.Z. Freedman and K. Skenderis, How to go with an RG ow, JHEP 08 (2001) [24] M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B [25] L. Girardello, M. Petrini, M. Porrati and A. Za aroni, The supergravity dual of N = 1 super Yang-Mills theory, Nucl. Phys. B 569 (2000) 451 [hep-th/9909047] [INSPIRE]. [26] D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrodinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE]. [27] M. Guica, K. Skenderis, M. Taylor and B.C. van Rees, Holography for Schrodinger backgrounds, JHEP 02 (2011) 056 [arXiv:1008.1991] [INSPIRE]. [28] R.N. Caldeira Costa and M. Taylor, Holography for chiral scale-invariant models, JHEP 02 (2011) 082 [arXiv:1010.4800] [INSPIRE]. perturbations, JHEP 02 (2015) 015 [arXiv:1410.6530] [INSPIRE]. avors, JHEP 08 (2015) 014 [arXiv:1505.07697] [INSPIRE]. HJEP01(28)4 mean curvature ow, Class. Quant. Grav. 34 (2017) 125005 [arXiv:1612.04373] [INSPIRE]. entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE]. theories on the three-sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE]. PoS(CONFINEMENTVIII)039 [J. Phys. A 42 (2009) 304005] [arXiv:0811.3824] [INSPIRE]. [11] E. Witten , Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 ( 1998 ) 253 [12] A. Lewkowycz and J. Maldacena , Generalized gravitational entropy , JHEP 08 ( 2013 ) 090 [29] M.P. Hertzberg and F. Wilczek , Some calculable contributions to entanglement entropy , Phys. Rev. Lett . 106 ( 2011 ) 050404 [arXiv: 1007 .0993] [INSPIRE]. [32] P.V. Buividovich and M.I. Polikarpov , Numerical study of entanglement entropy in SU(2) lattice gauge theory, Nucl . Phys. B 802 ( 2008 ) 458 [arXiv: 0802 .4247] [INSPIRE]. [33] E. Itou , K. Nagata , Y. Nakagawa , A. Nakamura and V.I. Zakharov , Entanglement in [35] H. Casini , M. Huerta and R.C. Myers , Towards a derivation of holographic entanglement [36] D.L. Ja eris, I.R. Klebanov , S.S. Pufu and B.R. Safdi , Towards the F-theorem: N = 2 eld [38] S. Aoki , T. Iritani , M. Nozaki , T. Numasawa , N. Shiba and H. Tasaki , On the de nition of


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Marika Taylor, William Woodhead. Non-conformal entanglement entropy, Journal of High Energy Physics, 2018, 4, DOI: 10.1007/JHEP01(2018)004